Details:

A linear code over a prime field can be described by a binomial ideal in a polynomial ring given as the sum of a toric ideal and a nonprime ideal. A Groebner basis for such an ideal can be read off from a systematic generator matrix of the corresponding code. In this paper, a similar result will be presented for linear codes over $\GF(4)$. To this end, the extented alphabet $\GF(4)$ is dealt with by enlarging the polynomial ring.